Inhomogeneous random graphs, isolated vertices, and Poisson approximation

TL;DR

This paper provides criteria for when the number of isolated vertices and other features in inhomogeneous random graphs can be approximated by a Poisson distribution, using Stein's method for Poisson approximation.

Contribution

It introduces general criteria for Poisson approximation of various graph features in inhomogeneous random graphs using Stein's method.

Findings

Poisson approximation criteria for isolated vertices

Extension to vertices of fixed degree and components

Effective use of Stein's method for Poisson approximation

Abstract

Consider a graph on randomly scattered points in an arbitrary space, with two points $x,y$ connected with probability $\phi(x,y)$. Suppose the number of points is large but the mean number of isolated points is $O(1)$. We give general criteria for the latter to be approximately Poisson distributed. More generally, we consider the number of vertices of fixed degree, the number of components of fixed order, and the number of edges. We use a general result on Poisson approximation by Stein's method for a set of points selected from a Poisson point process. This method also gives a good Poisson approximation for U-statistics of a Poisson process.